MAT3613

Topics List

Date	Sections	Topics	Prerequisite Skills	Student Learning Outcomes
Week I	Ahmad and Ambrosetti 2014, Chaps. 1, 2, 3	Order of Differential Equations	Integration techniques.	Explain the basic notion of the order of a differential equation.
Week I	Ahmad and Ambrosetti 2014, Chaps. 1, 2, 3	Solutions of Differential Equations	Integration techniques.	Explain the basic notion of solutions of differential equations.
Week I	Ahmad and Ambrosetti 2014, Chaps. 1, 2, 3	Initial Value Problem (IVP)	Integration techniques.	Explain the basic notion of the initial values problem.
Week I	Ahmad and Ambrosetti 2014, Chaps. 1, 2, 3	Cauchy Problem	Integration techniques.	Explain the Cauchy Problem Explain the basic notion of existence and uniqueness of a solution to the Cauchy Problem.
Week I	Ahmad and Ambrosetti 2014, Chaps. 1, 2, 3	Separation of Variables (1st Order)	Integration techniques.	Determine separable differential equations of the first order. Apply direct methods to evaluate exact solutions of separable differential equations of the first order.
Week II	Ahmad and Ambrosetti 2014, Chaps. 1 and 3	Homogeneous Differential Equations (1st Order)	Integration techniques.	Determine homogeneous differential equations of the first order. Apply direct methods to evaluate exact solutions of homogeneous differential equations of the first order (substitutions). Use some differential equations as mathematical models in biology, population dynamics, mechanics and electrical circuit theory problems.
Week II	Ahmad and Ambrosetti 2014, Chaps. 1 and 3	Linear Differential Equations (1st Order)	Integration techniques.	Determine linear differential equations of the first order. Apply direct methods to evaluate exact solutions of linear differential equations of the first order (substitutions, integrating factor method). Use some differential equations as mathematical models in biology, population dynamics, mechanics and electrical circuit theory problems.
Week III	Ahmad and Ambrosetti 2014, Ch. 3	Bernoulli Equations (1st Order)	Integration techniques: - Partial Derivatives - Linear Differential Equations (1st Order)	Determine Bernoulli of the first order. Apply direct methods to evaluate exact solutions of Bernoulli of the first order.
Week III	Ahmad and Ambrosetti 2014, Ch. 3	Exact Differential Equations (1st Order)	The integrating factor for exact equations. Integration techniques: - Partial Derivatives - Linear Differential Equations (1st Order)	Determine Exact Differential Equations of the first order. Apply direct methods to evaluate exact solutions of Exact Differential Equations of the first order. Use the integrating factor technique for exact equations.
Week IV	Ahmad and Ambrosetti 2014, Chaps. 1-3	Overview of the solutions methods discussed so far (Chapters 1-3).	Integration techniques: - Partial Derivatives First-order differential equations: - Separation of Variables (1st Order) - Homogeneous Differential Equations (1st Order) - Linear Differential Equations (1st Order) - Bernoulli Equations (1st Order) - Exact Differential Equations (1st Order)	Determine the type of different classes of differential equations of the first order: separable, linear, homogeneous, Bernoulli, exact. Use direct methods to solve first order differential equations solved and not solved for the first derivative.
Week V	Ahmad and Ambrosetti 2014, Ch. 5	Wronskian	Linear Algebra - Linear Independence. - Determinant.	Linear dependence and independence of functions. Showing linear independence of two functions using the Wronskian. Showing linear independence of two solutions of Linear Second-Order ODEs using the Wronskian.
Week VI	Ahmad and Ambrosetti 2014, Ch. 5	Reduction of the Order	Wronskian. Algebraic Equations. Determinant.	Apply of the reduction of the order technique for second-order ODEs with a given solution.
Week VI	Ahmad and Ambrosetti 2014, Ch. 5	Linear Differential Equations (2nd and Higher Order)	Wronskian. Algebraic Equations. Determinant.	Determine linear classes of differential equations of the second and higher order Determine non-linear classes of differential equations of the second and higher order
Week VI	Ahmad and Ambrosetti 2014, Ch. 5	Abel’s Theorem	Wronskian. Algebraic Equations. Determinant.	Determine Wronskian for a second-order ODE with 2 given solutions.
Week VI	Ahmad and Ambrosetti 2014, Ch. 5	Fundamental Solutions	Wronskian. Algebraic Equations. Determinant.	Determine fundamental solutions.
Week VI	Ahmad and Ambrosetti 2014, Ch. 5	Homogeneous Equations	Wronskian. Algebraic Equations. Determinant.	Determine homogeneous classes of differential equations of the second and higher order. Determine non-homogeneous classes of differential equations of the second and higher order.
Week VI	Ahmad and Ambrosetti 2014, Ch. 5	Variation of Parameters	Wronskian. Algebraic Equations. Determinant.	Apply of the variation of parameters technique for second-order ODEs.
Week VII	Ahmad and Ambrosetti 2014, Ch. 5	Variation of Parameters (continued)	Variation of parameters. Method of undetermined coefficients.	Apply variation of parameters technique for second-order ODEs.
Week VII	Ahmad and Ambrosetti 2014, Ch. 5	Method of Undetermined Coefficients	Variation of parameters. Method of undetermined coefficients.	Apply method of undetermined coefficients technique for second-order ODEs.
Week VIII		SPRING BREAK
Week IX		Preparation for remote instruction.
Week X	Ahmad and Ambrosetti 2014, Ch. 5	Higher order ODEs.	Methods for higher-order ODEs. Variation of parameters. Method of undetermined coefficients.	Apply variation of parameters and method of undetermined coefficients techniques for higher-order ODEs
Week XI	Ahmad and Ambrosetti 2014, Chaps. 5, 6, 10	Overview of the solutions methods for second and higher order differential equations. Collect HOMEWORK # 2 (extended deadline)	Direct methods for second and higher-order ODEs.	Evaluate the exact solutions of important classes of differential equations such as second order differential equations as well as some higher order differential equations.
Week XII	Ahmad and Ambrosetti 2014, Ch. 11	MIDTERM EXAM # 2: Second and higher-order ODEs Laplace transform. Definition. Main properties. HOMEWORK # 3 – L-transform. Applications of L-transform for ODES and systems of ODEs: Due at the beginning of Week XV	Improper integrals with infinite limits.	Definition and main properties of the Laplace transform.
Week XIII	Ahmad and Ambrosetti 2014, Ch. 11	Theorem(s) for inverse L- transforms	Derivatives of functions of complex variables.	Apply the theorem(s) for inverse L-transform.
Week XIV	Ahmad and Ambrosetti 2014, Ch. 11	Applications of L-transform to ODEs. Applications of L-transform to systems of ODEs.	Properties of the L- transform and inverse L-transform.	Apply the Laplace transform as solution technique.
Week XV	Ahmad and Ambrosetti 2014	Applications of L-transform to ODEs and systems of ODEs. Overview of the solutions methods discussed.	Solutions methods discussed.	Apply the L-transform. Apply all solutions methods discussed.
Week XVI	Ahmad and Ambrosetti 2014	Collect HOMEWORK # 3 Overview of the solutions methods discussed.	Solutions methods discussed.	Apply all solutions methods discussed.